Page 50 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

GGrraade11,, University Preparraattioionn
1. evaluate powers with rational exponents, simplify expressions containing exponents, and describe properties of exponential functions represented in a variety of ways;
2. make connections between the numeric, graphical, and algebraic representations of exponential functions;
3. identify and represent exponential functions, and solve problems involving exponential functions, including problems arising from real-world applications.
1. Representing Exponential Functions
2. Connecting Graphs and Equations of Exponential Functions
B. EXPONENTIAL FUNCTIONS OVERALL EXPECTATIONS
By the end of this course, students will:
THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics
SPECIFIC EXPECTATIONS
By the end of this course, students will:
1.1 graph, with and without technology, an expo- nential relation, given its equation in the form y = ax (a > 0, a ≠ 1), define this relation as the function f(x) = ax, and explain why it is a function
1.2 determine, through investigation using a variety of tools (e.g., calculator, paper and pencil, graphing technology) and strategies (e.g., patterning; finding values from a graph;
interpreting the exponent laws), the value of m
a power with a rational exponent (i.e., x n ,
where x > 0 and m and n are integers) Sample problem: The exponent laws suggest 11
that 42 x 42 = 41. What value would you
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assign to 42 ? What value would you assign 1
to 27 3 ? Explain your reasoning. Extend your
reasoning to make a generalization about the
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meaning of xn, where x > 0 and n is a natural number.
1.3 simplify algebraic expressions containing
integer and rational exponents [e.g.,
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(x3) ÷ (x 2 ), (x6y3) 3 ], and evaluate numeric
expressions containing integer and rational exponents and rational bases
–3 3 12 120
[e.g., 2 , (–6) , 4 , 1.01 ]
1.4 determine, through investigation, and describe key properties relating to domain and range, intercepts, increasing/decreasing intervals, and asymptotes (e.g., the domain is the set of real numbers; the range is the
set of positive real numbers; the function either increases or decreases throughout its domain) for exponential functions represented in a variety of ways [e.g., tables of values, mapping diagrams, graphs, equations of the formf(x)=ax (a>0,a≠1),function machines]
Sample problem: Graph f(x) = 2x, g(x) = 3x, and h(x) = 0.5x on the same set of axes. Make comparisons between the graphs, and explain the relationship between the y-intercepts.
By the end of this course, students will:
2.1 distinguish exponential functions from linear and quadratic functions by making compar- isons in a variety of ways (e.g., comparing rates of change using finite differences in tables of values; identifying a constant ratio in a table of values; inspecting graphs; compar- ing equations)
Sample problem: Explain in a variety of ways how you can distinguish the exponential function f(x) = 2x from the quadratic function f(x) = x2 and the linear function f(x) = 2x.
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